Demographics | Physical health | Hearing | Cognition | Mobility and balance | Quality of life | Mental health | Social participation and support | Views on hearing loss |
Logistic regression and other statistical methods require complete cases. In our case, with 40+ variables of interest, dropping participants with any missing values leads to a sample size of 200+ cases instead of 500+. There is loss of power, and estimates from models will likely be biased. The goal of missing data imputation is to restore variance of the sample to make it better reflect characteristics of the population (i.e., give accurate estimates of standard errors, CI’s, and p-values).
Say there are three variables ‘A’, ‘B’, and ‘C’, each with some missing values. Choose variable ‘A’ to start filling in. Fill in missing values in ‘B’ and ‘C’ with temporary values, such as the mean.
Predict missing ‘A’, using A ~ B + C, adding a random component. Move on to predicting missing ‘B’, using B ~ A + C and including predicted ‘A’, again adding a random component. Move on to predicting missing ‘C’, using C ~ A + B and predicted ‘B’. This is one cycle.
Choose the number of cycles, say 10 iterations. At the end of 10 iterations, an “imputed” dataset is formed with complete cases. Choose the number of imputed datasets to form (the same analysis will be conducted on each slightly different dataset, and the results will be pooled). Choose which variables should be used as predictors of other variables. Choose which method to use for predicting missing values, e.g., predictive mean matching. For derived variables such as summed scores, consider whether to impute components first and then calculate derived scores, or impute derived scores. Consider whether there should be an order of which variables to start imputing first.
Check whether the imputed data “converge” (the means and SEs stabilize after a few iterations, and are similar across imputed datasets). Check the distributions and characteristics of imputed data (e.g. integer-only values, floors and ceilings, plausibility), and their similarity to observed data.
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## 0 | 00004444444444444444444444444444444444444999999999999999999999999999+193
## 2 | 22222222222222222222222222222222222222222277777777777777777777777777+16
## 4 | 000044444999993333333888888888888
## 6 | 222222222222222666666666666666666111111111111111111155555555555
## 8 | 0000000004444888833337777
## 10 | 221199
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Participants with <26% missing data (n=509):
Participants with >42% missing data (n=18):
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## 0 | 00000000000000000000000000000000000000000000000000000000000000000000+152
## 2 | 33333333333333333333333333333333333333333333333333333333333333333333+50
## 4 | 555555555555555555555555555555555555555555555
## 6 | 88888888888888888888888888888888888888888888888888888888888
## 8 | 1111111111111111111111111
## 10 | 444444444444
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## 90 | 99
Participants with <20% missing data (n=509):
## # A tibble: 44 × 3
## variable n_miss pct_miss
## <chr> <int> <num>
## 1 Gender_bin 0 0
## 2 Age 2 0.393
## 3 Retired_bin 0 0
## 4 Volunteer 0 0
## 5 Lives_alone_bin 0 0
## 6 Phys_health_rating 7 1.38
## 7 Multimorbidity_score 0 0
## 8 Subj_vision_loss_bin 18 3.54
## 9 Subj_vision_rating_aided 1 0.196
## 10 Subj_hearing_loss_bin 36 7.07
## 11 any_hearing_device 0 0
## 12 study2_HA 0 0
## 13 Subj_hearing_rating_aided 159 31.2
## 14 Subj_hearing_rating_unaided 8 1.57
## 15 Tinnitus_past_wk_bin 0 0
## 16 ssq_speech 46 9.04
## 17 ssq_spatial 51 10.0
## 18 ssq_qualities 3 0.589
## 19 SIM_mean 12 2.36
## 20 Emocheq_mean 8 1.57
## 21 hhies_emo_total 0 0
## 22 hhies_soc_total 0 0
## 23 positive_SCI_bin 0 0
## 24 CSRQ_mean 0 0
## 25 Mobility_needs_bin 0 0
## 26 ABC_mean 0 0
## 27 SWLS_mean 0 0
## 28 WHOQOL_overall_qol 0 0
## 29 WHOQOL_health_qol 0 0
## 30 WHOQOL_Dom1_phys 0 0
## 31 WHOQOL_Dom2_psy 0 0
## 32 WHOQOL_Dom3_soc 1 0.196
## 33 WHOQOL_Dom4_env 0 0
## 34 WHO_money 0 0
## 35 PHQ4_mean 1 0.196
## 36 Lonely_bin 1 0.196
## 37 Social_network_index 1 0.196
## 38 Soc_part_freq 1 0.196
## 39 Soc_part_types 1 0.196
## 40 Connections_total 9 1.77
## 41 Motivate_mean 5 0.982
## 42 HA.Purchase 78 15.3
## 43 PTA4_better_ear 73 14.3
## 44 PTA4_asym 74 14.5
Participants with >39%% missing data (n=18):
## # A tibble: 43 × 3
## variable n_miss pct_miss
## <chr> <int> <num>
## 1 Gender_bin 2 11.1
## 2 Age 3 16.7
## 3 Retired_bin 2 11.1
## 4 Volunteer 2 11.1
## 5 Lives_alone_bin 2 11.1
## 6 Phys_health_rating 4 22.2
## 7 Multimorbidity_score 0 0
## 8 Subj_vision_loss_bin 4 22.2
## 9 Subj_vision_rating_aided 4 22.2
## 10 Subj_hearing_loss_bin 5 27.8
## 11 any_hearing_device 0 0
## 12 study2_HA 0 0
## 13 Subj_hearing_rating_aided 10 55.6
## 14 Subj_hearing_rating_unaided 5 27.8
## 15 Tinnitus_past_wk_bin 0 0
## 16 ssq_speech 15 83.3
## 17 ssq_spatial 13 72.2
## 18 ssq_qualities 8 44.4
## 19 SIM_mean 13 72.2
## 20 Emocheq_mean 13 72.2
## 21 hhies_emo_total 15 83.3
## 22 hhies_soc_total 15 83.3
## 23 positive_SCI_bin 15 83.3
## 24 CSRQ_mean 17 94.4
## 25 Mobility_needs_bin 17 94.4
## 26 ABC_mean 17 94.4
## 27 SWLS_mean 17 94.4
## 28 WHOQOL_overall_qol 18 100
## 29 WHOQOL_health_qol 18 100
## 30 WHOQOL_Dom1_phys 18 100
## 31 WHOQOL_Dom2_psy 18 100
## 32 WHOQOL_Dom3_soc 18 100
## 33 WHOQOL_Dom4_env 18 100
## 34 WHO_money 18 100
## 35 PHQ4_mean 18 100
## 36 Lonely_bin 18 100
## 37 Social_network_index 18 100
## 38 Soc_part_freq 18 100
## 39 Soc_part_types 18 100
## 40 Connections_total 18 100
## 41 Motivate_mean 18 100
## 42 HA.Purchase 5 27.8
## 43 PTA4_better_ear 5 27.8
Complete cases out of n=527 (none of 44 variables missing):
## [1] 232
## [1] 44.02277
## # A tibble: 44 × 3
## variable n_miss pct_miss
## <chr> <int> <num>
## 1 Gender_bin 2 100
## 2 Age 2 100
## 3 Retired_bin 2 100
## 4 Volunteer 2 100
## 5 Lives_alone_bin 2 100
## 6 Phys_health_rating 2 100
## 7 Multimorbidity_score 0 0
## 8 Subj_vision_loss_bin 2 100
## 9 Subj_vision_rating_aided 2 100
## 10 Subj_hearing_loss_bin 2 100
## 11 any_hearing_device 0 0
## 12 study2_HA 0 0
## 13 Subj_hearing_rating_aided 2 100
## 14 Subj_hearing_rating_unaided 2 100
## 15 Tinnitus_past_wk_bin 0 0
## 16 ssq_speech 2 100
## 17 ssq_spatial 2 100
## 18 ssq_qualities 2 100
## 19 SIM_mean 2 100
## 20 Emocheq_mean 2 100
## 21 hhies_emo_total 2 100
## 22 hhies_soc_total 2 100
## 23 positive_SCI_bin 2 100
## 24 CSRQ_mean 2 100
## 25 Mobility_needs_bin 2 100
## 26 ABC_mean 2 100
## 27 SWLS_mean 2 100
## 28 WHOQOL_overall_qol 2 100
## 29 WHOQOL_health_qol 2 100
## 30 WHOQOL_Dom1_phys 2 100
## 31 WHOQOL_Dom2_psy 2 100
## 32 WHOQOL_Dom3_soc 2 100
## 33 WHOQOL_Dom4_env 2 100
## 34 WHO_money 2 100
## 35 PHQ4_mean 2 100
## 36 Lonely_bin 2 100
## 37 Social_network_index 2 100
## 38 Soc_part_freq 2 100
## 39 Soc_part_types 2 100
## 40 Connections_total 2 100
## 41 Motivate_mean 2 100
## 42 HA.Purchase 2 100
## 43 PTA4_better_ear 2 100
## 44 PTA4_asym 2 100
## # A tibble: 44 × 3
## variable n_miss pct_miss
## <chr> <int> <num>
## 1 Gender_bin 0 0
## 2 Age 1 6.25
## 3 Retired_bin 0 0
## 4 Volunteer 0 0
## 5 Lives_alone_bin 0 0
## 6 Phys_health_rating 2 12.5
## 7 Multimorbidity_score 0 0
## 8 Subj_vision_loss_bin 2 12.5
## 9 Subj_vision_rating_aided 2 12.5
## 10 Subj_hearing_loss_bin 3 18.8
## 11 any_hearing_device 0 0
## 12 study2_HA 0 0
## 13 Subj_hearing_rating_aided 8 50
## 14 Subj_hearing_rating_unaided 3 18.8
## 15 Tinnitus_past_wk_bin 0 0
## 16 ssq_speech 13 81.2
## 17 ssq_spatial 11 68.8
## 18 ssq_qualities 6 37.5
## 19 SIM_mean 11 68.8
## 20 Emocheq_mean 11 68.8
## 21 hhies_emo_total 13 81.2
## 22 hhies_soc_total 13 81.2
## 23 positive_SCI_bin 13 81.2
## 24 CSRQ_mean 15 93.8
## 25 Mobility_needs_bin 15 93.8
## 26 ABC_mean 15 93.8
## 27 SWLS_mean 15 93.8
## 28 WHOQOL_overall_qol 16 100
## 29 WHOQOL_health_qol 16 100
## 30 WHOQOL_Dom1_phys 16 100
## 31 WHOQOL_Dom2_psy 16 100
## 32 WHOQOL_Dom3_soc 16 100
## 33 WHOQOL_Dom4_env 16 100
## 34 WHO_money 16 100
## 35 PHQ4_mean 16 100
## 36 Lonely_bin 16 100
## 37 Social_network_index 16 100
## 38 Soc_part_freq 16 100
## 39 Soc_part_types 16 100
## 40 Connections_total 16 100
## 41 Motivate_mean 16 100
## 42 HA.Purchase 3 18.8
## 43 PTA4_better_ear 3 18.8
## 44 PTA4_asym 3 18.8
Method:
* predictors that correlate 0.5 or better; if no such predictors, adjust
threshold to 0.4, then 0.3
* total of 5 imputed datasets, with a maximum of 10 iterations per
dataset
Note: Not filtered by hearing aid candidacy according to threshold(s) at 2kHz.
Non-hearing aid users = 166
Hearing aid users = 80
Significant predictors: Subj_hearing_rating_aided,
Subj_hearing_rating_unaided, SWLS_mean, PTA4_better_ear
“Borderline”: SSQ15i_mean, SWLS_mean
##
## Call:
## glm(formula = formula_adopt, family = "binomial", data = compdata_ad)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -32.741259 1452.817468 -0.023 0.98202
## Gender_bin 1.595429 1.040222 1.534 0.12509
## Age -0.064929 0.063040 -1.030 0.30303
## Retired_bin 0.266908 1.063635 0.251 0.80186
## Volunteer_rec 0.898904 0.539164 1.667 0.09547 .
## Lives_alone_bin -1.202422 1.152395 -1.043 0.29676
## Phys_health_rating 0.396070 0.782182 0.506 0.61260
## Multimorbidity_score 0.282669 0.246895 1.145 0.25225
## Subj_vision_rating_aided -0.652282 0.608955 -1.071 0.28410
## Subj_hearing_loss_bin 18.953453 1452.771718 0.013 0.98959
## Subj_hearing_rating_aided 4.410219 0.934386 4.720 0.00000236 ***
## Subj_hearing_rating_unaided -3.678818 0.939617 -3.915 0.00009032 ***
## Tinnitus_past_wk_bin -0.405308 0.763624 -0.531 0.59558
## SSQ15i_mean 0.687075 0.375327 1.831 0.06716 .
## SIM_mean 0.131242 0.269486 0.487 0.62625
## Emocheq_mean -0.469772 0.602305 -0.780 0.43542
## HHIES_total -0.007491 0.076414 -0.098 0.92191
## positive_SCI_bin 0.003053 1.276869 0.002 0.99809
## CSRQ_mean 0.595525 1.230241 0.484 0.62834
## Mobility_needs_bin -0.806893 2.067784 -0.390 0.69637
## ABC_mean -0.045911 0.056297 -0.816 0.41478
## SWLS_mean 1.264797 0.666152 1.899 0.05761 .
## WHOQOL_overall_qol -0.849883 0.879340 -0.967 0.33379
## WHOQOL_health_qol 0.033379 0.671457 0.050 0.96035
## WHO_money 0.977226 0.627806 1.557 0.11957
## PHQ4_mean 0.812446 1.016468 0.799 0.42413
## Lonely_bin 2.348070 1.591187 1.476 0.14003
## Social_network_index -0.276936 0.401326 -0.690 0.49016
## Soc_part_freq 0.403448 0.662299 0.609 0.54242
## Soc_part_types 0.114114 0.411978 0.277 0.78179
## Motivate_mean -0.882616 0.603671 -1.462 0.14372
## PTA4_better_ear 0.164242 0.061936 2.652 0.00801 **
## PTA4_asym 0.065309 0.043998 1.484 0.13771
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 310.319 on 245 degrees of freedom
## Residual deviance: 70.645 on 213 degrees of freedom
## AIC: 136.64
##
## Number of Fisher Scoring iterations: 19
Significant predictors: Volunteering, Subj_hearing_rating_aided, Subj_hearing_rating_unaided, SSQ15i_mean, ABC_mean (balance), PHQ4_mean (anxiety & depression), PTA4_better_ear
“Borderline”: Subj_hearing_loss_bin, SWLS_mean, PTA4_asym
## term estimate std.error statistic df
## 1 (Intercept) -0.8069314818 0.347688012 -2.32084930 221.99196
## 2 Gender_bin 0.0213103466 0.035588740 0.59879463 181.13383
## 3 Age 0.0030946174 0.002458405 1.25879050 147.67164
## 4 Retired_bin -0.0329322640 0.034810440 -0.94604561 373.95398
## 5 Volunteer_rec 0.0384696492 0.021014378 1.83063465 102.98698
## 6 Lives_alone_bin -0.0421223132 0.040621309 -1.03695114 108.10358
## 7 Phys_health_rating 0.0048974761 0.024822393 0.19730072 479.64343
## 8 Multimorbidity_score 0.0010202011 0.007497715 0.13606826 221.41715
## 9 Subj_vision_rating_aided -0.0014353804 0.022066377 -0.06504831 364.36256
## 10 Subj_hearing_loss_bin 0.0844530941 0.055600558 1.51892529 32.82621
## 11 Subj_hearing_rating_aided 0.1689107116 0.026451476 6.38568202 14.66008
## 12 Subj_hearing_rating_unaided -0.1857980295 0.028057215 -6.62211243 95.78959
## 13 Tinnitus_past_wk_bin -0.0233440561 0.034242844 -0.68172072 195.88066
## 14 SSQ15i_mean 0.0340272696 0.014147564 2.40516816 113.40805
## 15 SIM_mean -0.0003149891 0.007511506 -0.04193422 455.56254
## 16 Emocheq_mean -0.0030424740 0.019886497 -0.15299195 406.62382
## 17 HHIES_total 0.0033770670 0.002734797 1.23485128 346.69664
## 18 positive_SCI_bin 0.0448430673 0.053323627 0.84096057 352.29619
## 19 CSRQ_mean -0.0170251042 0.046529860 -0.36589631 346.04220
## 20 Mobility_needs_bin -0.0385328494 0.080872256 -0.47646562 212.81316
## 21 ABC_mean -0.0026130143 0.001687901 -1.54808532 76.55505
## 22 SWLS_mean 0.0282781959 0.018999204 1.48838842 125.90802
## 23 WHOQOL_overall_qol 0.0026437000 0.037230582 0.07100883 53.51686
## 24 WHOQOL_health_qol 0.0231610629 0.023214441 0.99770065 101.98604
## 25 WHO_money 0.0202992253 0.021711638 0.93494673 31.69368
## 26 PHQ4_mean 0.1057386289 0.039162194 2.70001802 323.42657
## 27 Lonely_bin -0.0222851838 0.044412033 -0.50178257 264.69183
## 28 Social_network_index -0.0043010248 0.013630313 -0.31554850 170.09644
## 29 Soc_part_freq 0.0111070032 0.025880134 0.42917101 348.56966
## 30 Soc_part_types -0.0055868731 0.012025513 -0.46458501 306.53098
## 31 Motivate_mean 0.0139793267 0.018237954 0.76649643 417.41902
## 32 PTA4_better_ear 0.0062904258 0.002357231 2.66856575 30.77204
## 33 PTA4_asym 0.0028221170 0.001372063 2.05684188 268.72822
## p.value
## 1 0.021202335223233
## 2 0.550057952873636
## 3 0.210092352772850
## 4 0.344736207353346
## 5 0.070046775338566
## 6 0.302072561198147
## 7 0.843675794718204
## 8 0.891890945202822
## 9 0.948171212636370
## 10 0.138355038748211
## 11 0.000013642781195
## 12 0.000000002051981
## 13 0.496220804627005
## 14 0.017782907611612
## 15 0.966569508762473
## 16 0.878480511227692
## 17 0.217722090998015
## 18 0.400940446495265
## 19 0.714666027216855
## 20 0.634231525301153
## 21 0.125726032185143
## 22 0.139148870536759
## 23 0.943655425709040
## 24 0.320785858070180
## 25 0.356885099935214
## 26 0.007298342454854
## 27 0.616237592331074
## 28 0.752732052336673
## 29 0.668063909362946
## 30 0.642558482578352
## 31 0.443814231253736
## 32 0.012044452103685
## 33 0.040666695793303
Significant predictors: Lives_alone, Phys_health_rating, Subj_hearing_rating_unaided, HHIES_total, Social_network_index
“Borderline”: Gender, Multimorbidity, Tinnitus_past_wk_bin, WHOQOL_health_qol
##
## Call:
## glm(formula = formula_subj, family = "binomial", data = compdata_subj)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 7.20205 11.32644 0.636 0.524866
## Gender_bin 2.31782 1.24407 1.863 0.062448 .
## Age 0.09764 0.07394 1.321 0.186665
## Retired_bin 0.50181 1.03681 0.484 0.628389
## Volunteer_rec -0.25097 0.52291 -0.480 0.631258
## Lives_alone_bin 3.55977 1.40448 2.535 0.011258 *
## Phys_health_rating 2.49657 1.06599 2.342 0.019180 *
## Multimorbidity_score 0.48840 0.26459 1.846 0.064915 .
## Subj_vision_rating_aided 0.77191 0.65064 1.186 0.235469
## Subj_hearing_rating_aided 1.26440 0.83753 1.510 0.131127
## Subj_hearing_rating_unaided -4.80248 1.26743 -3.789 0.000151 ***
## Tinnitus_past_wk_bin 1.72993 0.98404 1.758 0.078749 .
## SSQ15i_mean 0.08561 0.64183 0.133 0.893891
## SIM_mean -0.14577 0.29619 -0.492 0.622615
## Emocheq_mean -0.71311 0.77177 -0.924 0.355491
## HHIES_total 0.35817 0.13188 2.716 0.006609 **
## positive_SCI_bin 0.74498 1.47506 0.505 0.613525
## CSRQ_mean -1.20147 1.28087 -0.938 0.348239
## Mobility_needs_bin 3.76606 3.13412 1.202 0.229506
## ABC_mean -0.03983 0.09533 -0.418 0.676069
## SWLS_mean -0.25840 0.52024 -0.497 0.619401
## WHOQOL_overall_qol -0.83342 0.92690 -0.899 0.368577
## WHOQOL_health_qol -1.81784 0.94566 -1.922 0.054570 .
## WHO_money 0.55103 0.55792 0.988 0.323319
## PHQ4_mean -1.27155 1.46151 -0.870 0.384289
## Lonely_bin -2.72142 1.68101 -1.619 0.105464
## Social_network_index 0.82708 0.35702 2.317 0.020523 *
## Soc_part_freq -0.96243 0.80039 -1.202 0.229188
## Soc_part_types -0.68527 0.44169 -1.551 0.120793
## Motivate_mean -0.53730 0.67082 -0.801 0.423160
## PTA4_better_ear 0.10468 0.07061 1.483 0.138186
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 317.160 on 245 degrees of freedom
## Residual deviance: 59.256 on 215 degrees of freedom
## AIC: 121.26
##
## Number of Fisher Scoring iterations: 9
Significant predictors: Subj_vision_rating_aided, Subj_hearing_rating_unaided, Tinnitus_past_wk_bin, HHIES_total, PTA4_better_ear
“Borderline”: Emocheq_mean
## term estimate std.error statistic df
## 1 (Intercept) 0.7099298119 0.427590265 1.66030396 19.52395
## 2 Gender_bin 0.0379102617 0.040586547 0.93405979 32.64963
## 3 Age 0.0020794484 0.002419673 0.85939233 161.92456
## 4 Retired_bin 0.0348938908 0.034727332 1.00479619 324.90659
## 5 Volunteer_rec -0.0146770924 0.020653726 -0.71062684 116.29450
## 6 Lives_alone_bin -0.0013293405 0.044594500 -0.02980952 34.86516
## 7 Phys_health_rating 0.0235154439 0.025161770 0.93457036 340.21224
## 8 Multimorbidity_score 0.0080939353 0.008192223 0.98800231 49.56151
## 9 Subj_vision_rating_aided 0.0523675355 0.021821709 2.39979071 338.27996
## 10 Subj_hearing_rating_aided 0.0146752705 0.028726996 0.51085294 10.83156
## 11 Subj_hearing_rating_unaided -0.2235336361 0.031317784 -7.13759434 17.56214
## 12 Tinnitus_past_wk_bin 0.1323700886 0.034484599 3.83852772 107.15377
## 13 SSQ15i_mean -0.0060635793 0.013426847 -0.45160112 194.15172
## 14 SIM_mean 0.0018748161 0.008739200 0.21452950 41.40915
## 15 Emocheq_mean -0.0374399307 0.020435921 -1.83206480 179.37327
## 16 HHIES_total 0.0067603410 0.003038584 2.22483270 51.07739
## 17 positive_SCI_bin 0.0137610662 0.052693748 0.26115178 369.14409
## 18 CSRQ_mean -0.0094620895 0.050446754 -0.18756588 68.34825
## 19 Mobility_needs_bin 0.1013811405 0.078915752 1.28467560 279.61267
## 20 ABC_mean 0.0015764352 0.001573811 1.00166737 201.20502
## 21 SWLS_mean -0.0106246817 0.019549489 -0.54347618 69.90539
## 22 WHOQOL_overall_qol -0.0005018867 0.039673266 -0.01265050 28.85071
## 23 WHOQOL_health_qol -0.0344836751 0.021676616 -1.59082370 274.34265
## 24 WHO_money 0.0227800828 0.021335015 1.06773219 32.93603
## 25 PHQ4_mean -0.0491196776 0.045483446 -1.07994626 35.29514
## 26 Lonely_bin -0.0673929501 0.047807373 -1.40967693 66.62459
## 27 Social_network_index 0.0044710001 0.013797135 0.32405280 117.95596
## 28 Soc_part_freq -0.0050696977 0.026897683 -0.18848083 138.87030
## 29 Soc_part_types -0.0048532026 0.012875520 -0.37693254 78.48268
## 30 Motivate_mean -0.0254518356 0.018183903 -1.39969049 366.91367
## 31 PTA4_better_ear 0.0113530427 0.002067921 5.49007436 59.07400
## p.value
## 1 0.1128295622527
## 2 0.3571300944617
## 3 0.3913945954169
## 4 0.3157427253118
## 5 0.4787382246249
## 6 0.9763889570944
## 7 0.3506726072902
## 8 0.3279506080679
## 9 0.0169446192010
## 10 0.6197048583372
## 11 0.0000013784760
## 12 0.0002097205527
## 13 0.6520604244966
## 14 0.8311871534972
## 15 0.0686001338052
## 16 0.0305366796645
## 17 0.7941210951214
## 18 0.8517726964903
## 19 0.1999689228947
## 20 0.3177073846139
## 21 0.5885309756394
## 22 0.9899937056994
## 23 0.1128008802340
## 24 0.2934052665623
## 25 0.2874907755576
## 26 0.1632861827149
## 27 0.7464719606668
## 28 0.8507748952463
## 29 0.7072409846903
## 30 0.1624509260467
## 31 0.0000008886159